chaytr'r 3 71

~haptrr 3

D£V£LOPMENT Of TROPOSPH£RIC Dt:LA Y MODELS

3.0 Introduction

Neutral atmosphere being non-dispersive in nature, the only way to mitigate the

tropospheric delay is to estimate it from the refractivity profiles, accomplishing a priori

knowledge of which to incorporate in operational model is rather difficult. Then it becomes

essential to develop conventional models relating the Tropospheric delay (both ZHD and

ZWD) with easily measurable atmospheric parameters at the surface. Though a typical mean

value of - 2.3 m for ZHD and 20 to 40 cm for ZWD can generally be taken as a "rule of

thumb", the day-to-day variability in atmospheric parameters can introduce a variation up to

±15 cm about this mean value, as far as tropospheric range error is concerned. Because of

this, conventional models [Saastamoinen, 1972; Davis et al., 1985; lfadis, 1986; Mendes

and Langley, 1998] for ZHD and ZWD are developed in terms of atmospheric parameters at

the surface, such as surface pressure (Ps), surface temperature (Ts) and surface water vapor

pressure (es).

Saastamoinen [1972] developed a simple relation between ZHD and Ps for different

latitudes taking into account the altitude variation also, which is fairly accurate up to 0.4 km

for different seasons. Later Davis et al. [1985} and Elgered [1991] proposed similar models

for ZHD with a slightly different value for the relevant coefficients. All these models are

developed under the assumption that the dry atmosphere is in hydrostatic equilibrium and

follows the equation of state. However, since water vapor need not be in hydrostatic

equilibrium and is heterogeneously distributed in the atmosphere, similar models were not

attempted for ZWD until lfadis [1986J who developed simple empirical relation between

ZWD and ps. Ts and es. Later Mendes and Langley [1998] also proposed a simple empirical

relation for ZWD using es and Ts. A slightly different model for ZHD and ZWD are evolved

by Hopfield [1971], following a different approach based on refractivity profile.

A brief outline of these models is summarized in Chapter 1 Section 1.8. It would be

worth in this context to note that all these empirical models are developed based on

meteorological data from European and North American continents [Satirapod and

c'hayter 3 72

Chalermwattanachai, 2005]. The accuracies of these models, assessed by comparing with

the true estimates obtained from ray-tracing, are typically < 3 mm for ZHD [Janes et al.

1991], and < 5 cm for ZWD [Askne and Nordius, 1987; lanes et al., 1991; Elgered, 1992]

with a maximum deviation extending up to - 8 cm during abnormal weather events

[Elgered, 1993; lchikawa, 1995]. However, no such studies are reported till date over the

tropical/subtropical region. Taking note of this fact, empirical models applicable for the

tropical region are developed based on the meteorological data from different locations

situated at distinct climatic conditions over the Indian subcontinent. These models are

validated by comparing with the true range error (estimated by ray-tracing).

3.1 Site-Specific Surface Models for Indian Subcontinent

The relationship between zenith tropospheric delay and the atmospheric parameters at

surface are examined for different stations located at different climatic zones over the Indian

subcontinent. A simple linear relationship is established for ZHD in terms of surface

pressure through regression analysis. The true value of ZHD is estimated by applying the

integral (eg. 2.39) to the monthly mean atmospheric models described in Section 2.6 and an

empirical linear model connecting ZHD with Ps is developed in the following form

ZHD = aD' Ps (3.1)

Figure 3.1 shows typical scatter plot of ZHD estimated through ray tracing with Ps (in hPa)

for eight among the eighteen stations selected for this analysis (Table 2.1). These stations

represent the major climatic zones over the Indian region and also have the location in

proximity to major airports, considered in the GAGAN project. These stations are grossly

referred to as the Range Integrity and Monitoring (RIM) stations. The regression analysis is

performed keeping the value of intercept as zero and the resulting slope (aD) and correlation

coefficient are listed in the respective scatter plots. For all these stations, the value of

correlation coefficient lies between 0.82 and 0.99. The standard error of aD for each station

is < 0.024% which corresponds to about 0.05 cm in delay. The standard deviation (SD) of

the correlation also is listed in these plots. The values of coefficients aD thus estimated for

all the stations are presented in column 2 of Table 3.1.

Following a similar approach linear models are established for ZWD on a monthly

mean basis for different stations with different surface humidity parameters such as surface

--e ->,

= -~ Q ~ '. -= -rIl 0 1-0

"C >,

== ..c -.• I:: ~

N

2.32

2.31

2.30

2.29

2.28

2.27

"0.0.002281:3.3"10"

R = 0.82 SO = 0.002

la) TRIVANORUM

999 1000 1001 1002 1003 1004 1005

2.11.--------------------, 110 = Q.00228 :!:3.2'10·'

2.10 R=0.97; SO = O.0014m

2.09

2.08

2.07

2.06

Ic) 2.05

904 906 908

2.32 "0 = 0.00229 .! 4.4'10"

2.31 R = 0.98 SO = 0.00216

2.30

• 2.29 ••

• 2.28 • •

2.27

(e) 2.26

992 996 1000

2.27 a, = 0.00%3 .! 4.73'10"

2.26 R = 0.99 SO = 0.00226

2.25

2.24 •• •

2.23 •

2.22

Ig) 2.21

964 988 972

BANGALORE

910 912 914

• ••

•

•

AHMEDABAD

1004 1008 1012

•• •

DELHI

978 980 984 968

2.34 110 = 0.00231 .! 5.4'10"

2.33 R = 0.87 SO = 0.00381

2.32

2.31

• 2.30

2.29

(b)

•

PORTBLAIR 2.28 +--"T""~_r~--,---<'-.-___,--r__~r__,.......j

996 997 998 999 1000 1ilO1 1002 1ilOJ 1004

2.34,------------------.," 0.00229:!: 3.5'10·'

2.33 R " 0.99; SO = 0.0017

• •• 2.32

• 2.31 • •

2.30

2.29 • 2.28

Id) KOLKATA

996 1000 1ilO4 1008 1012 1016

Surface Pressure (hPa)

73

Figure 3.1: Scatter plots along with the regression line showing the variation of ZHD with Ps for different stations located at different climate zones over the Indian subcontinent and adjoining oceanic region

chayt(f 3 74

Table 3.1: Coefficients of Site-Specific Surface Models for Different Indian Stations

Dry Wet

Station ZHD (m) = aJJxPS ZWD (m) = aw><es ZWD (m) = bw'Xps ZWD (m) = CwX PW

aDxlO·3 aw bw Cw xlO·3

Trivandrum 2.280 ± 3.3E-4 0.01199 ± 1.5E-4 16.595 ± 0.233 6.45 ± 5.6E-3 (0.82; 0.002) (0.77; 0.0217) (0.72; 0.023) (0.99; 0.0015)

Portblair 2.310 ± 5.4E-4 0.01212 ± 1.8E-4 16.753 ± 0.255 6.45 ± 4.3E-3 (0.87; 0.0038) (0.93; 0.025) (0.91; 0.025) (0.99; 0.0011)

Bangalore 2.21\0 ± 3.2E-4 0.0122 ± 2.48E-4 16.703 ± 0.384 6.53±7.IE-3 (0.97: 0.0(14) (0.86; 0.025) (0.8 I; 0.0278) (0.99; 0.0013)

Mangalol"c 2.280 ± 3.4E-4 0.0113 ± 2.2E-4 15.628± 0.316 6.35 ± 7.7E-3 (0.89; D.OO 17) (0.85; 0.030) (0.85; 0.032) (0.99; 0.00) 9)

Chcnnai 2.290 ± 2.7E-4 O.O[ 11 ± 3. lE-4 15.409± 0.443 6.33 ± 7.8E-3 (0.98; 0.0013) (0.58; 0.042) (0.54; 0.043) (0.99; 0.0019)

Goa 2.280 ± 6.6E-4 0.0[09± 2.9E-4 15.047± 0.40 6.34±7.1E-3 (0.81; 0.0(32) (0.92; 0.040) (0.92; 0.040) (0.99; 0.0017)

Hyderabad 2.230 ± 3.0E-4 0.0142 ± 2.5E-4 19.577± 0.413 6.42 ± 5.6E-3 (0.98; 0.0(14) (0.94; 0.026) (0.90; 0.(30) (0.99; 0.0013)

Vishakhapatnam 2.270 ± 3.4E-4 0.QI07± 2.8E-4 15.477± 0.87 6.39 ± 7.8E-3 (0.99: 0.0017) (0.81; 0.039) (0.41; 0.089) (0.98; 0.0200)

Mumbai 2.290 ± 3.7E-4 0.0117± 3.8E-4 16.267± 0.56 6.33 ± 5.5E-3 (0.97; 0.(017) (0.92; 0.049) (0.90; 0.051) (0.99; 0.0013)

Kolkata 2.290 ± 3.5E-4 0.0119 ± 3.01E-4 16.43 ± 0.427 6.43 ± 9.2E-3 (0.99; 0.0(17) (0.94; 0.0408) (0.94; 0.0418) (0.99; 0.0023)

Ahmedabad 2.290 ± 4.4E-4 0.0119 ± 2.27E-4 16.627 ± 0.315 6.43 ± l.IE-2 (0.98; 0.0022) (0.97; 0.0245) (0.97; 0.0243) (0.99; 0.0022)

Bhopal 2.160 ± 6.IE-4 0.0112± 2.5E-4 15.394± 0.38 6.37 ± 3.7E-3 (0.97; 0.0(28) (0.96; 0.023) (0.95; 0.025) (0.99; 0.0060)

Guwahati 2.290 ± 3.0E-4 0.0[27 ± 2.7E-4 17.520 ± 0.399 6.46 ± I.IE-2 (0.99: 0.0(15) (0.97; 0.0338) (0.97; 0.(36) (0.99; 0.0027)

Jodhpur 2.230 ± 7.1 E-4 0.0122± 2.0E-4 16.847± 0.30 6.34 ± 0.2E-3 (0.98: 0.0(32) (0.98; 0.018) (0.98; 0.019) (0.99; 0.0028)

Lucknow 2.260 ± 5.2E-4 0.0 126± 4.5E-4 17.418±0.66 6.25 ± 0.3E-3 (0.98: 0.0025) (0.95; 0.048) (0.95; 0.051) (0.99; 0.0069)

Delhi 2.300 ± 4.7E-4 0.0128 ± 2.3E-4 17.539 ± 0.373 6.44 ± [.4E-2 (0.99: 0.0(23) (0.99; 0.023) (0.98; 0.0274) (0.99; 0.0024)

Patiala 2.200 ± 9.4E-4 0.0101± 8.4E-4 13.929± 1.15 6.00 ± 1.13E-1 (0.95; 0.0044) (0.70; 0.084) (0.70; 0.083) (0.99; 0.0069)

Srinagar 2.280 ± 6.5E-4 0.0118 ± 1.5E-4 15.934±0.[72 6.61 ± 2.5E-2 (0.97; 0.0(27) (0.99; 0.0093) (0.99; 0.0078) (0.99; 0.0027)

75

water vapor pressure (es in hPa), surface water vapor density (Ps, in kg m-3) and the

columnar integrated water vapor (Is in kg m-2) also known as Precipitable Water vapor (PW

in mm). As a typical example the scatter plots of ZWD estimated through ray tracing, with

es for the eight RIM stations are presented in Figure 3.2. A linear relationship was sought

analogous to that of the dry component through regression analysis keeping the value of

intercept as zero in the following form

ZWD = aw x es

ZWD =bw xPs

ZWD=cwxPW (3.2)

values of the coefficients thus estimated are presented in columns 3,4 and 5 of Table 3.1 for

all the eighteen stations. The quantities shown inside the parenthesis in each cell are the

correlation coefficient and its standard deviation, respectively. A perusal of Figure 3.1 along

with Figure 3.2 shows that the deviation of points from the mean line is relatively large in

the latter case, the effect of which is also observable in the standard error associated with the

regression coefficients. This shows that, in general, the uncertainty in the prediction of ZWD

is larger than that of ZHD. The regression coefficients connecting ZWD with es, Ps and PW

are also examined by removing the imposed condition of "zero intercept". The values of

regression coefficient are rearranged in a more convenient form as shown below and the

coefficients ao, bo, Co, eo, PG, and PW u are estimated.

ZWD = ao x (es - eo)

ZWD = ho x (Ps - Po)

ZWD = Co x (PW - PWo)

for es > eo

for Ps> Po

forPW > PWo (3.3)

The values of these coefficients along with their respective standard error are summarized in

Table 3.2. The respective values of the correlation coefficient and its standard deviation are

presented within the parenthesis just below the corresponding coefficients. The values of ao,

and bo are marginally greater than corresponding values of aw, and bw for all the 18 stations

except Chennai, Hyderabad, Vishakhapatnam, Bhopal, Jodhpur, Patiala and Srinagar where

the values of es, Ps and PW are relatively small. The values of Co and Cw for different

stations are more or less comparable. As these models are developed for each station

separately based on the monthly mean atmospheric models and surface meteorological

parameters from respective stations, they are termed as "Site-Specific surface models" in

chaytir 3

0.6,------------------, la) TRIVANDRUM

0.4

0.3

0.2

0.1

25 26 27 28 29 30 31

0.5,------------------,

0.4

0.3

0.2

~ 0.1 ~ ..-~

(c) BANGALORE

1"'\ 0.0 +-~~_._~----r~---.-~..__~_.__I ~ 14 16 1B 20 22 24 ..... ~ > O.S

;;> (e)

-s 0.4 ..... I=l ~ N 0.3

0.2

0.1

AHMEDABAD

•• • -- ..

-

0.0+--~--r-~__,.-~-r_--,_~-1

10 15 20 25 30 35

0.5,-------------------, DELHI Ig)

•• 0.4

0.3

0.2 - _.

0.1

0.5,.....---------------~ (b) PORTBLAIR

0.4

0.3

0.2

0.1

0.0 +-~.-~.--.__,.-~-_.__~""T"'--r-'T"""~._,.......j 23 24 25 28 27 28 29 30 31 32

0.5..-----------------~

Cd)

0.4

, 0.3

• • • • • 0.2 -- -0.1

KOLKATA

15 20 25 30 35 40

0.5.,.------------------~

(11

0.4

• 0.3

0.2

0.1

•

-

--­,,-

GUWAHATI 0.0 +-_r----r-_-..,-_,._-...--..,--,--~-

10 15 20 25 30 35

0.5,.....---------------~

(hI SRINAGAR

0.4

0.3

0.2

0.1

0.0 0.0 +-...-..---,_-..-~__r_~T""""~ _ _,._~...__.........,,._,.___l

5 10 15 20 25 30 35 4 6 8 10 12 14 16 18 20 22

76

Surface water vapor pressure (hPa) Figure 3.2: Scatter plots along with the regression line showing the variation of ZWD with es for different stations located at different climate zones over the Indian subcontinent and adjoining oceanic region

chaytr'r 3 77

Table 3.2: The Regression Coefficients Connecting ZWD with eST Ps and PW, Removing

the Imposed Condition of "Zero Intercept"

Wet range error

ZWD (m) ;;; ao X (es-eo) ZWD (m) ;:: bo X (Ps-Po) ZWD (m) ;:: Co X (PW- PWo) Station

(ao±Aao) eo±Mo boiMo

(Pr,)±f1f}u) (co±&:o) PWu±f1PWo xlO-3 X 10.3 X 10.3

Trivandrum 15.11 ± 2.7

7.82 ± 0.08 21.39 ± 4.5

5.98 ± 4.5 6.35 ± 0.05

0.19±2e-3 (0.77;(1.019) (0.71 ;0.02) (0.99;0.001 )

Portblair 19.17±1.5

10.59 ± 0.04 24.47 ± 2.3

6.58 ± 2.3 6.46 ± 0.03

0.07 ± 1.6e-3 (0.93;0.018) (0.91;0.021) (0.99;0.00 I)

Bangalore 14.13 ± 1.8

2.80 ± 0.04 17.65 ± 2.7

0.806 ± 2.7 6.56 ± 0.04

O. [74 ± 1.4e-3 (0.86;0.024 ) (0.81 ;0.028) (0.99:0.00 I)

Mangalore 17.34 ± 2.3

9.9 ± 0.06 25.69 ± 3.4

8.03 ± 3.4 6.47 ± 0.04

1.00 + 2.2e-3 (0.85;0.027) (0.85;0.027) (0.99;0.002)

Chcnnai 11.95 ± 3.6

2.07 ± 0.10 15.99 ± 5.4

0.73 ± 5.4 6.30 ± 0.05

0.27 ± 2.3e-3 (0.58;0.043) (0.54;0.044) (O.99;0.002)

Goa 18.81 ± 1.7

11.89 ± 0.05 25.69 ± 2.4

8.5 ± 2.35 6.42 ± 0.03

0.567 ± 1.3e-3 (0.92;0.030) (0.92;0.030) (0.99;0.002)

Hyderabad 15.53 ± 1.2

[.77 ± 0.03 21.22 ± 2.1

1.18 ± 2.1 6.40 + 0.02

0.[5 ± l.le-3 (0.94;0.026) (0.9! ;0.030) (0.99;0.00 I)

Vishakhapatnam 11.45 ± 1.8

1.95 ± 0.05 12.3 ± 5.94

5.43 ± 5.9 6.85 ± 0.30

3.6 ± J.5e-2 (0.81 ;0.039) (0.40;0.090) (0.9R;0.OJ9)

Mumbai [6.71 ± 1.6

8.14 ± 0.04 23.10 ± 2.4

5.78 ± 2.4 6.33 ± 0.02

0.07 ± 1.0e-3 (0.92;0.041 ) (0.91 ;0.044) (0.99;0.00 I)

Kolkata 13.80 ± 1.0

3.92 ± 0.03 [9.81 ± 1.5

3.53 ± 1.5 6.34 ± 0.02

0.79 ± 1.0e-3 (0.94;0.038) (0.94;0.038) (0.99;0.002)

Ahmcdabad 12.37 ± 0.6

0.86 ± 0.01 16.98 ± 0.9

0.35 ± 0.8 6.37 ± 0.03

0.45 ± l.le-3 (0.97;0.025) (0.97:0.U25) (0.99:0.002)

BhopaJ 11.08 ± 0.7

0.13±0.01 15.16±1.1

0.22 ± 1.0 6.13±0.08

1.42 ± 2.7e-3 (0.96;0.023 ) (0.95;0.Q25) (0.99;O.005)

Guwahati 15.69 ± 0.8

4.96 ± 0.02 22.44 ± 1.5

4.19 ± 1.2 6.34 ± 0.02

1.05 ± 0.8e-3 (0.97;0.027) (0.97 ;0.027) (0.99;0.00 I)

Jodhpur [2.12 ± 0.5

0.04 ± 0.01 17.42 ± 0.8

0.50 ± 0.7 6.22 ± 0.Q2

0.82 ± 0.ge-3 (0.98;0.018) (0.98;0.019) (0.99;0.002)

Lucknow [5.59 ± 1.0

4.59 ± 0.Q2 21.96 ± 1.6

3.60 ± 1.6 6.05 ± 0.05

1.72 ± 2. le-3 (0.95;0.041 ) (0.95:0.044) (0.99:0.005)

Delhi 14.30 ± 0.5

2.34 ± 0.01 20.20 ± 0.7

2.14±0.8 6.34 ± 0.01

0.76 ± 0.7e-3 (0.98;0.019) (0.98;0.022) (0.99;O.002)

Patiala 9.90 ± 2.2

0.43 ± 0.05 14.11 ± 3.2

0.21 ±3.2 5.50 ± 0.20

4.15 ± 6.2e-3 (0.70;0.086) (0.70;0.085) (0.99;0.016 )

Srinagar 11.70 ± 0.4

0.15 ± 0.01 16.58 ± 0.4

0.39 ± 0.4 6.39 ± 0.03

0.87 ± 0.6e-3 (0.98;(J.OO9) (0.99;0.007) (0.99;0.001)

78

further discussions. However, the accuracy of these models in general depends on the

accuracy of the surface atmospheric parameters [Back and Doerflinger, 2000].

3.2 Adaptation of Hopfield Model for the Indian Region

The surface models provide integrated range error all the way from top of the

atmosphere to the surface. Whenever the prediction of tropospheric range error is required

from an elevated platform, a priori knowledge of the altitude dependences of the relevant

coefficients is to be accounted through an appropriate correction factor [Baby et al., 1988].

Taking this factor into account, the Hopfield [1969] model (which addressed the altitude

dependence in range error) is adapted for the Indian region. As detailed in Section 1.5.4, in

this model the integral effect of refractive index parameter above the user altitude alone is

considered. The altitude variation of refractivity is modeled through the semi-analytical

relations connecting the surface refractivity with the characteristic height parameter.

Correspondingly two characteristic height parameters evolve, one for the hydrostatic

component (hD) and the other for the non-hydrostatic component (hw). The altitude profile

of dry component of refractivity (ND) when h $ ho can be expressed as,

(3.4)

where ho (in km) is the "characteristic height" for dry refractivity profile, hs is the altitude of

the station above MSL and Nso is the value of ND at the surface (where h = hs). The equation

for dry component is obtained by analytically solving the gas law equations applicable for

the atmosphere [Hop field, 1969]. As the temperature lapse rate varies with geographic

latitude as well as to certain extent with altitude also, " also varies accordingly. Even

though, no such analytical form can be arrived in the case of the wet component (Nw),

Hapfield [1969] in his formulation assumed that the water vapor also follows a similar law

with same value of "but different value of characteristic height (hw). With this assumption

the altitude variation of wet component of refractivity (Nw), when h $ hw, is modeled as

(3.5)

where hw (in km) is the 'characteristic height' for wet refractivity profile and Nsw is the

value of Nw at the surface. In the above formulations, it is implicitly assumed that the effect

79

of ND above hD and that of Nw above hw is insignificant. As the zenith delay is obtained by

integrating the atmospheric refractivity along the altitude between hs and top of the

atmosphere, ZHD and ZWD can be estimated by analytically integrating eg. (3.4) and eq.

(3.5) respectively, replacing the upper limits by the respective characteristic heights. If we

consider the altitude profiles of ND and Nw starting from the surface, for any location hs can

be treated as zero. The expression for zenith range errors can then be written as.

hIJ h ZHD = fNDdh = NSD __ D_

o (17+ 1) (3.6a)

and

hw hw ZWD= fN .... eldh=Nsw--

o (17 + 1) (3.6b)

The above equations are strictly valid only when the integrals are taken up to the respective

characteristics height.

If we consider the mean sea level values of the dry and wet components of the

refractivity (NOD and Now, respectively), then NSD and Nsw at the surface (h = hs) can be

written in terms of their MSL values as

(3.7a)

and

[hw -hs)1I

Nsw =Now hw (3.7b)

Then eq. (3.6a) and eg. (3.6b) can be rewritten accordingly in terms of NOD and Now.

Following a similar approach, eq. (3.7a) and eq. (3.7b) both can be extrapolated for any

given altitude by replacing hs with that altitude. In case if one is interested in using the

surface values NSD and Nsw, same procedure can be followed by replacing NOD and Now in

eq. (3.7a) and eq. (3.7b), respectively, by NSD and Nsw. In this case hs should be replaced by

the height of the location above the surface (hp).

However, this method requires a priori knowledge of the characteristic height

parameters and atmospheric refractive index at the surface. The characteristic height is

derived by equating the numerical integral of refractive index profile to the theoretically

estimated value of integrated refraction effect given in eq. (3.6a) and eq. (3.6b). In this

80

process f3 decides the value of 7] and hence the denominator of the eq. (3.6). Even though

using different temperature lapse rate will change the value of 7], the value of hD estimated

through eq. (3.6a) using the new lapse rate also will change correspondingly in such a way

that the ratio of ho/( TJ+ 1) remains the same. This is true in case of hw also. However, in the

present study the temperature lapse rate is taken as 6.7 Klkm as was done by Hopfield

[1971] and tre values of hD and hw derived for all the stations are based "n this lapse rate.

Implementation of this model, however, involves modeling of the characteristic height

parameters for different stations. Substituting the true range error estimated by integrating

the refractivity profiles obtained from the monthly mean atmospheric models in eq. (3.6a)

and eq. (3.6b) respectively; the dry and wet characteristic heights are estimated at each

station for different months. The month-to-month variation of ho estimated for 00:00 UTC

and 12:00 UTe (corresponding to 05:30 1ST and 17:30 1ST, respectively) for all the

eighteen stations used for the present study are presented in Figure 3.3(a-r). Similarly,

Figure 3.4(a-r) shows the month-to-month variation of hw for these stations. The vertical

lines in these figures show the respective standard deviations in each month (representing

the day-to-day variability) estimated by applying the error propagation fonnula [Ku, 1966]

appropriately. While the month-to-month variation (seasonal) of ho is small for the tropical

stations « ±0.5 km), it is large for the high-latitude stations (- 5 km). While the mean value

of hw varies between - 11 km to - 14 km for the tropical stations it varies in the range 8 km

to 16 km for the high latitude stations. Significant annual variation in hw is observable for

the high-latitude stations (with amplitude as large as - 9 km). A distinct morning to evening

difference is observable in the values of ho for all these stations with the evening values

consistently being larger than the corresponding morning values, even though such a trend

does not emerge for hw. However, in most of the cases the evening values of hw are larger

than the morning values. The annual mean values of ho and hw for the eighteen stations are

presented in Table 3.3 along with the respective standard deviations, within parenthesis

(indicating the extent of mean monthly variations). While the observed mean values of ho

and hw are quite large and fairly constant for the near equatorial stations (Trivandrum,

Portblair and Bangalore), a prominent annual variation is observable for the other stations

particularly in case of ho. The characteristic height for different months can be used as a

look up table for each station for estimating the range error employing eq. 3.6(a) and eq.

3.6(b) by incorporating the respective values of surface refractivity.

chaytt'T 3 81

Table 3.3: Mean Value of hD and hw along with the Maximum Month-to-month Variability (within parenthesis)

Stations hD (km) hw (km)

Trivandrum 44.610 (0.39) 13.32 (0.88)

Portblair 44.580 (0.25) 13.34 (1.10)

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86

3.2.1 Dependence of hD and hw on Surface Temperature

The morning to evening difference in characteristic heights could be attributed mainly

to the corresponding changes in the atmospheric temperature. While this dependence is

distinctly observable for ho• the dependence is rather weak for hw. Then it would be possible

to generate an empirical relation for hD in terms of surface temperature which will be useful

for the application even on a day-to-day basis. Though a systematic morning to evening

difference is not observable for hw. it would be convenient to treat this also analogous to ho

keeping in mind that eq. (3.5) was also proposed on these lines. The only additional input

required in this case would be the value of surface temperature. Once the necessary inputs

for the model viz. surface pressure, water vapor partial pressure and surface temperature are

fed, the parameters Nso. Nsw, ho• and hw can directly be generated using the appropriate

relationships. With this motivation empirical models for ho and hw are developed based on

linear regression. Figure 3.5 shows a scatter plot depicting the temperature dependence of ho

for different RIM stations along with the best fit line obtained through the regression

analysis in the following form

ho = hDO + go Ts (3.8)

The values of hoo and go for all the 18 stations are presented in Table 3.4. The values of hoo

are around 40 km and are fairly constant for all the stations. The values of go, the

temperature gradient of hD is around 0.14 for an these stations.

Following a similar procedure, the empirical relation connecting hw with Ts is

established based on the observed dependencies as depicted in Figure 3.6. As can be seen

from this figure. the scatter of points is relatively large, and the correlation of hw with Ts is

inferior. This could be attributed to the fact that water vapor is not well mixed in the

atmosphere and its altitude variation is not exactly similar to that of the other major gas­

constituents in the atmosphere, not withstanding the fact that higher atmospheric

temperature is conducive even for holding more water vapor and extending the profile to

higher altitude. Large-scale advection of water vapor during the monsoon period is another

factor responsible for the observed low correlation between hw and Ts.

Note that, in developing a quartic model for Nw. no theoretical justification was

provided by Hopfield for adopting such a relationship. It was just adapted analogous to that

for hydrostatic component mainly for the convenience in modeling. Keeping these factors

47~--------------------------------, R = 0.99 SO = 0.017 TRIVANORUM

44

43

42

(a) 41+-~--r-----.--r~r-~~--~-.--~~

22 24 26 28 30 32 34

47~--------------------------------,

46

45

44

! 43

'-' .... .= 42 ~

R = 0.999; SO = 0.027 BANGALORE

.. ~ 41+-~~~r-~r-~~(C~I'-~.-~.-~r-~ .= 16 CJ

18 20 22 24 26 26 3D 32 .. ~ 47 .--R-:-0-.9-9-S-0--=-O-.O-24-2-5--------A-HM--EO-A-B-A-O-,

... ~ 46 U ~ ... 45 ~ .= U 44

to 43 Q

42

12 16 20 24 28 32 36 40 44

~.------------------------------, R ~ 0.99 SO = 0.031 DELHI

46

45

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47~------------------------------, R = 0.96 SO = 0.067 PORTBLAIR

46

45

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22 23 24 25 26 27 28 29 30 31

47 R=0.99; SO = 0.029 KOLKATA

46

45

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(d) 41+-~--.-~--.-~--r-----r--r-.~

10 15 20 25 30 3S

47.-----------------------------~ R = 0.999 SO = 0.023 GUWAHATI

46

45

44

43

42

10 12 14 16 16 20 22 l4 26 28 30

45,---------------------------, R = 0.999 SO = 0.063 SRINAGAR

44

43

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40

41+-~-r~-.~-.~(g~J-.~-, __ ~.-~ 39+-~r-~r-~~-(~~~)~~~.-~~~ 10 15 20 25 30 35 40 -6 0 6 10 15 20 25 30 35

Surface Temperature (OC)

87

Figure 3.5: Scatter plots along with best-fit regression line showing the surface temperature dependence of dry characteristic height (hD)

chayter 3 88

17 17 R = 0.21; SO = 0.88 TRIVANDRUM R = 0.36; SO = 1.006 PORTBLAIR

16 16

15 15 .. .. ... 14 r ... .. ; 14 ... ... 13 ... 13 ..

... ... 12 12 • • • • ... .. • 11 11 ... • 10 10

9 (a) (b)

9 22 24 26 28 30 32 22 23 24 25 26 27 28 29 30 31 32

17 17 R = 0.69; SO=1.17 BANGALORE R = 0.43; SO = 1.51 KOL.KATA

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Surface Temperature (OC) Figure 3.6: Scatter plots along with best-fit regression line showing the surface temperature dependence of wet characteristic height (hw)

89

Table 3.4: Empirical Relations Representing the Temperature Dependencies for the Dry and Wet Characteristic Height Parameters (hD and hw) for Different Stations

Model for hD Model for hw

Station Name ho = hoo + go Ts hw = hwo + gw Ts

hOG (km) go hwo(km) gw Trivandrum 40.86{0.03) O.l40JO.001) 11.58 (1.75) 0.065 (0.064)

Portblair 40.74 (0.02) 0.146 (0.008) 7.355 (3.46) 0.229 (0.13) Bangalore 40.37 (0.03) 1.14610.001) 7.746 (1.25) 0.230 (0.051) Mangalore 40.46 (0.05) 0.141 (0.002) 14.42 (2.18) 0.063 (0.080)

Chennai 40.28 (0.04) 0.146 (0.001) 7.583 (2.84) 0.180 (0.101) Goa 40.33 (0.11) 0.145 (0.004) 10.23 (3.03) 0.068 (0.111)

Hyderabad 39.36 (0.02) 0.141 (6e-4) 7.792 (1.07) 0.311 (0.039) Vishakhapatnam 39.83 (0.06) 0.151 (0.002) 9.737 (3.42) 0.092 (0.125)

Mumbai 40.23 (0.17) 0.148 (0.006) 8.642 (3.17) 0.152{O.l15) Kolkata 40.28 (0.03) 0.147 (0.001) 9.457 (1.53) 0.131 (0.058)

Ahmedabad 40.36 (0.02) 0.143 (7e-4) 9.923 (1.00) 0.120 (0.035) Bhopal 37.92 (0.03) 0.144 (0.00l) 6.978 (1.15) 0.233 (0.044)

Guwahati 40.34 (0.02) 0.144 (ge-4) 8.730 (1.29) 0.198 (0.050) Jodhpur 39.59 (0.49) 0.121 (0.018) 12.10 (0.94) 0.072 (0.033)

Lucknow 39.75 (0.04) 0.146 (0.001) 7.371 (1.82) 0.244 (0.072) Delhi 40.52 (0.02) 0.144 (8e-4) 9.150 (0.7) 0.178 (0.028)

Patiala 38.41 (0.06) 0.140 (0.002) 8.312 (1.05) 0.076 (0.042)

Srinagar 40.14 (0.02) 0.144 (0.001) 11.50 (0.306) 0.046 (0.020)

into account in order to facilitate the modeling, an empirical relation is established for the

variations of hw with Ts similarly that for hD as,

hw = hwo + gw Ts (3.9)

The coefficients estimated through linear regression for each station relating hw to Ts is

included in Table 3.4. While for the dry characteristic height the value of intercept (hoo) is

fairly constant for all the stations (about 40.5 ± 0.24 km) with a mean slope of 0.144 ± 0.002

km/°C, for wet characteristic height the mean intercept (hwo) is about 9.43 km with a slope

of 0.14 kmf'c. The variability of the intercept and slope in the case of hw for different

stations is large compared to that of hD . However, this is not a major constraint because

even with these limitations the above relationship can provide fairly good prediction of hw.

These models for different stations are referred to as "Site-specific Hopfield model" in

further discussions.

Both the Site-Specific Surface model and Hopfield model estimate the zenith

tropospheric delay in terms of surface meteorological parameters. The essential difference

between these two models is that while the former provides this in terms of surface

90

hydrostatic pressure and water vapor partial pressure, the latter includes the effect of surface

temperature also.

3.3 A Unified Model for the Indian Subcontinent

3.3.1 Unified Surface Model

A close examination of empirical relations derived based on the surface model and

Hopfield model for the individual stations show that the variability of the coefficients

connecting the range error with meteorological parameters is rather small. This prompted

development of a Unified Model (each for surface model and Hopfield model) applicable for

all these stations which hence will be applicable for the entire region encompassing these

stations. Though such unified models could be inferior to the site-specific models, its

potential increases because of the fact that one simple equation will be sufficient for the

entire region. Following the same methodology adopted for individual stations, the relation

between surface meteorological parameters and dry component of range error is sought by

pooling together the data from all the eighteen stations. Figure 3.7a shows the scatter plot of

surface pressure with zenith dry range error, considering all the stations. A simple linear

relation is established based on linear regression (keeping zero intercept) and the slope is

estimated. This yields a simple linear relation termed as the "Unified Surface model" as in

the form

ZHD = (0.00228 ± 2.25xlO-6) x Ps (3.10)

with a correlation coefficient of 0.97. In eq. (3.10) Ps is expressed in hPa and ZHD is

obtained in meters. Following a similar approach, a linear relation connecting ZWD and es

also is established for the whole subcontinent. A scatter plot of ZWD, with the

corresponding values of es is shown in Figure 3.7b. The linear relation of the fonn

(3.11)

is established between the two through regression analysis and evaluated the coefficients Aa

and AI. The values of Aa and Al are, respectively, -0.01978 ± 0.0057 m and 0.01288 ±

2.4x1O-4 m hPa- l. As can be seen from Figure 3.7b, when es is very large (>30 hPa) the

range error estimated through the relation (shown by thin straight line) is consistently lower

than the true range error obtained from ray-tracing (marked points). Similarly a higher

altitude station like Srinagar has lower values of es compared to near sea-level stations

chayter 3 91

(Trivandrum and Portblair). In order to distinguish the values corresponding to Srinagar,

these are plotted in Figure 3.7b with a different symbol (V).

2.6

2.4

E 2.2 '-'

Cl :t N 2.0

1.8

(a) Slope = 0.00228! 1.3915591:-6 Corr. Coeff. = 0.97; er = 0.019

1.6 +-.....--......-,---r-,---"--r--,.....,--.---,.--....--.---l 900 920 940 960 980 1000 1020

Surface Pressure (P sI in mb

U.5

0.4

0.3 E Cl

~ 0.2

0.1

Second Order Polynomial R = 0.95 SO = 0.025 m

Linear Model (b) R = 0.968 SO = 0.027 m

5 10 15 20 25 30 35 ~~ (mb)

0.5..,..-----------------.,..,

0.4

E 0.3

C

~ 0.2

0.1

R = 0.999 SO = 0.0027

(c)

10 20 30 40 50 60 70 80

PW{mm)

Figure 3.7: Regression analysis showing the dependence of dry and wet components of tropospheric zenith range on relevant surface atmospheric parameters considering all the eighteen stations together for generating a unified model: Variation of dry range error with surface pressure (a), Variation of wet range error with surface water vapor partial pressure (b) and variation of wet range error with columnar (precipitable) water vapor (c)

A careful examination of Figure 3.7b shows that the points in this scatter plot are more

aligned to a non-linear increase of ZWD with es. Considering this aspect the applicability of

a second order polynomial is examined to relate ZWD with es, which yielded a relationship

of the form

chaytir 3 92

ZWD= (0.0391±0.013)+ (0.00636±0.OOl) xes +(1.58xl0-4 ±3.75xlO-s)xe/ (3.12)

The nonlinear variation of ZWD by this relationship is shown in Figure 3.7b with the thick

line.

3.3.2 Unified Model in terms ofPrecipitable Water Vapor

As seen in Section 3.1, a linear model relating PW with ZWD will be more appropriate

for predicting ZWD than similar models based on surface values of water vapor partial

pressure, or water vapor density at surface, as far as individual stations are concerned.

However, it should be noted in this context that it will not be possible to get values of PW at

any location whenever prediction of ZWD is required. On the contrary it will be easier to get

values of es or Ps from many places at almost all the time. Hence from the application point

of view such models (relating ZWD with es or Ps) will be more useful even though the error

associated with them will be comparatively larger. One of the ways for estimating PW is by

using a sun-photometer working on the principle of differential attenuation (of water vapor

and other atmospheric species at two optical wavelengths). This instrument operates only

during day time under clear sky conditions and requires regular calibration. Ground based

microwave radiometer, which can work in all weather condition (except heavy precipitation)

though is another potential instrument for measuring PW, is very expensive. The estimation

of PW from radiosonde is not only expensive (recurring expenditure when ever launched)

but also has a poor temporal and spatial resolution (considering the time taken for each

sounding and the balloon drift). The satellite based microwave radiometers though are very

useful over the oceanic regions they will not be available quite frequently. A GPS network

could be a good source of PW especially over the land (not too expensive). This can be

operated in all weather conditions with a very good temporal resolution and does not require

any calibration. In this case, a Unified Model to estimate ZWD from PW will be useful.

Figure 3.7c shows a scatter plot of PW with true zenith wet delay (estimated from

refractivity profiles derived from radiosonde) clubbing all the eighteen stations. A linear

model was developed based on the regression analysis, which yielded a relationship of the

form

ZWD = (0.0064 ± 4.4xl 0-6) ~ PW (3.13)

chayter 3 93

with correlation coefficient close to unity. As can be seen directly fonn Figure 3.7c, this

model is significantly superior to the other three models. The mean absolute difference for

different stations is presented in the last column of Table 3.7 along with the corresponding

standard deviation. A similar model was also used by Bevis et al. [1992] for deriving ZWD

from ground based GPS measurements. In his global model, Bevis et al. [1992] evolved a

value of 0.00666 as the ratio of ZWDfPW which varies as much as 20% with location,

season and weather condition [Be vis et al., 1994].

3.3.3 Unified Hopfield Model (Characteristic Height Parameters)

Following a very similar approach a unified empirical model is developed for zenith

tropospheric delay based on the Hopfield method. This essentially involves deriving a

unified model for dry and wet characteristic height parameters in tenns of Ts by grouping

the data from all the eighteen stations. The temperature dependence of hD and hw are

established incorporating the data from all the 18 stations by linear regression which is of

the form

hD ;::: (40.209 ± 0.045) + (0.154 ± 0.002) >; Ts

hw = (10.474 ± 0.29) + (0.111 ± 0.01) >; Ts

(3.14a)

(3.14b)

A typical scatter plot of hD (in km) with Ts (in QC) and hw (in km) and Ts (in 0c)

obtained by combining all the stations is presented in Figure 3.8a and Figure 3.8b along with

the best fitted line represented by eq. 3.14a and eg. 3.14b, respectively. The correlation

coefficient for hD and hw are 0.98 and 0.56, respectively. Comparing the values of these

regression coefficients with those presented in Table 3.4 (site-specific surface model), it can

be seen that the present value of slope is slightly larger than the corresponding values in

Table 3.4 while the value of the intercept is slightly small.

3.4 Validation of Surface Models

3.4.1 Validation of Site-Specific Models by Comparing with Ray-Traced Values

The site-specific models are validated by comparing the model predictions with the

true range errors for the respective station obtained through ray tracing employing the

altitude profiles of refractivity. Sufficient care is taken to ensure that the daily

94

meteorological data have water vapor profile extending up to or beyond la km altitude, and

the temperature and pressure profiles extending up to at least 25 km. The comparison is

quantified by estimating the absolute difference between true and model predicted zenith

delays, a summary of which is presented in Table 3.5. The data for RIM stations are

highlighted with bold face. The mean of the absolute differences are presented along with

the standard deviations. While the mean value indicates the average deviation of the model,

the standard deviation indicates the amount of variability. The accuracy of surface model

based on Ps is comparable to that of the Hopfield model in predicting ZHD. The mean

deviation is lowest for Trivandrum and highest for Jodhpur.

48 18

R = 0.987; SD = 0.187 R=O.S6 SO'" 1.2

46 I~

44 14 - --2 :: ::. ::.

l!: 1> ~ 42 .::: 12 1>

1> 1>

1>

40 JO 1> 1> 11:

(a) (b) 38 8

10 20 3D 40 SO 0 IU 10 30 4U

T/C) T~(C)

50

Figure 3.8: Regression analysis showing the dependence of dry (hD) and wet (hw) characteristic heights on surface temperature CTs) considering all the eighteen stations together

The deviation of Site-Specific models, in case of ZWD, based on es and Ts is about 5-

6 cm for most of the stations. For a few stations like Vishakhapatnam, Mumbai, Jodhpur

and Lucknow the deviation is > 6 cm. Srinagar (being high altitude station) shows

minimum mean absolute difference about -2 cm for ZWD. Among all the RIM stations

considered, Kolkata shows maximum deviation of -6 cm. The best estimate of wet delay

comes from model based on Precipitable Water vapor. The mean absolute difference for

this model is -0.3 cm except at Patiala and Lucknow. By combining the surface models

based on Ps and the wet range error model based on PW, the best estimate of ZTD (with an

error <6 cm) can be achieved for almost all the stations. As an alternative (when PW is not

c'hayter 3 95

available) the surface models based on es or Hopfield model based on es and Ts (which are

more easily available) could be considered, even though the involved accuracies are rather

inferior. Note that, the accuracy achieved by these models also meets basic requirement of

satellite based navigation system discussed in Chapter 1.

Table 3.5: Mean Absolute Difference and the standard deviation of Site-specific Models (fl"om true zenith range error) for Different Stations (RTM stations in Bold), for the Hydrostatic and Non-hydrostatic Components

Hydrostatic delay (ZHD) Non-hydrostatic delay (ZWD)

Station Mean abs. Diff (cm) Mean abs. Diff (cm)

Surface mode Hopfield Surface model Hopfield Surface model based on Ps model based on es model based on PW

Trivandrum 0.17 (0.1) 0.15 (0.1) 5.2 (3.7) 5.1 (3.6) 0.17 (0.1) Portblair 0.36 (0.3) 0.49 (0.4) 4.5 (3.5) 4.6 (3.6) 0.16 (0.1) Bangalore 0.45 (0.2) 0.2 (0.2) 4.0 (3.0) 4.2 (3.0) 0.16 (0.1)

Mangalore 2.3 (1.9) 2.5 (2.5) 4.9 (3.6) 6.3 (4.3) 0.64 (0.3)

Chennai 4.5 (2.7) 4.4 (2.7) 6.1 (4.5) 5.9 (4.4) 0.6 (0.3)

Goa 2.4 (0.5) 1.5 (0.5) 5.4 (3.5) 5.7 (3.9) 0.5 (0.23)

Hyderabad 3.7 (2.0) 3.5 (2.0) 6.7 (4.6) 6.1 (4.4) 0.4 (0.26)

Vishakhapatnam 3.5 (2.6) 3.6 (2.7) 7.6(6.3) 7.5 (6.2) 0.3 (0.2)

Mumbai 3.18 (2.8) 3.12 (2.7) 6.9 (4.3) 7.5 (4.5) 0.51 (0.27)

Kolkata 0.35 (0.2) 0.32 (0.3) 6.0 (4.2) 6.4 (4.3) 0.21 (0.1)

Ahmedabad 0.34 (0.3) 0.33 (0.2) 5.2 (4.0) 5.3 (4.0) 0.24 (0.2)

Bhopal 1.25 (1.2) 1.35 (1.25) 6.5(5.1) 5.8 (4.6) 0.66 (0.3)

Guwahati 0.24 (0.2) 0.25 (0.2) 4.4 (3.3) 5.0 (3.6) 0.20 (0.1)

Jodhpur 11.2 (6.2) 10.0 (5.5) 9.5 (8.1) 9.4(8.1) 0.4 (0.3)

Lucknow 5.7 (5.7) 5.7 (5.8) 8.8 (12.9) 8.6 (12.7) 1.1 (1.3)

Delhi 0.27 (0.2) 0.27 (0.2) 4.0 (3.0) 4.5 (3.6) 0.20 (0.2)

Patiala 2.7 (2.3) 3.4 (2.6) 4.1 (3.0) 5.6 (3.8) 2.0 (0.4)

Srinagar 1.80 (1.7) . 1.09 (1.8) 2.0 (2.0) 1.9 (1.6) 0.19 (0.1)

3.4.2 Validation of Unified Models using Ray-Tracing

The accuracy of unified model is also examined by comparing the estimated delays

with those obtained by ray tracing as well as with other global models based on surface

parameters [Saastamoinen, 1972; Hopfield, 1971; Ifadis, 1986; Mendes and Langley, 1998].

The mean of the absolute difference of the deviations for each station is presented in Table

3.6. These values are larger than those for site-specific surface models presented in Table

3.5, especially for Trivandrum and Ahmedabad. For the case of ZWD, the comparison of

chayter 3 96

Table 3.6: Mean Absolute Difference and the standard deviation of Unified Model and Global Model from True Zenith Range Error for Different Stations, for the Hydrostatic Component

Hydrostatic delay

Station Mean abs. Diff (cm)

Code SAAS Surface Model Hopfie1d

(1972)

Ps Ps model

Trivandrum 2.66 (0.2) 1.01 (0.2) 1.24 (0.27) Portblair 2.57 (0.4) 0.94 (0.4) 1.10 (0.40) Bangalore 1.03 (0.3) 0.45 (0.2) 0.30 (0.20) Mangalore 2.33 (2.13) 2.29 (1.88) 2.87 (2.66)

Chennai 4.09 (2.5) 3.84 (2.35) 5.15 (2.83) Goa 2.10 (0.47) 2.40 (0.47) 1.00 (0.43)

Hyderabad 2.28 (2.45) 2.23 (2.25) 3.08 (2.75) Vishakhapatnam 4.70 (2.80) 4.33 (2.75) 5.90J2.84)

Mumbai 2.86 (2.50) 2.79 (2.26) 3.39 (3.15) Kolkata 0.84 (0.4) 0.97 (0.4) 0.80 (0.4)

Ahmedabad 0.70 (0.4) 1.10 (0.4) 0.99 (0.5) Bhopal 12.10 0.73) 11.70 (1.73) 13.25 (1.7 1)

Guwahati 0.85 (0.6) 0.79 (0.5) 0.93 (0.8) ]odhpur 16.02 (6.68) 15.66 (6.66) 17.25 (6.83)

Lucknow 7.40 (6.26) 7.08 (6.19) 8.45 (6.45) Delhi 1.90 (0.3) 0.28 (0.2) 0.47 (0.25) Patiala 7.76 (3.11) 7.44 (3.03) 8.81 (3.40)

Srina2ar 2.40 (2.6) 3.20 (2.0) 2.90 (2.0) All stations

2.90 (3.60) 2.90 (3.4) 2.98 (4.15) together

both linear (eq. 3.11) and second order (eq. 3.12) models are presented in Table 3.7. In case

of ZWD the deviation of unified model is comparable to that of Site-Specific model. Among

different models suggested for ZWD, the second order polynomial Unified model yields a

better prediction. Considering all the eighteen stations together the mean of the absolute

difference for SAAS and Unified model from the true value is - 2.9 cm. In the case of wet

delay, the Unified models developed in the present analysis are much superior to the

currently available models [Ifadis, 1986 and Mendes and Langley, 1998]. The mean absolute

difference of the Unified model based on es varies from about 4 cm to 9 cm whereas that in

case of other two global models is in the range 5 cm to 15 cm. This study indicates the

region specific models (for tropics) are superior to the available global models developed

97

Table 3.7: Mean Absolute Difference and the standard deviation of Unified Models and Global Models from True Zenith Range Error for Different Stations, for the Non­hydrostatic Component

Non-hydrostatic delay Mean abs. Diff (cm)

Mendes Ifadis Surface 2na order

Hopfield Surface

(1997) (1986) Model- Model-

Model Model-

es es PW

Trivandrum 8.15 9.18 5.1 4.8 4.98 0.17 (4.7) (4.8) (3.6) (3.7) (3.5) (0.17)

Portblair 7.51 8.49 4.5 4.4 4.40 0.16 (5.2) (5.3) (3.5) (3.5) (3.3) (0.13)

BangaIore 8.12 8.08 4.3 4.9 4.20 0.57 (3.8) (3.8) (2.9) (2.8) (2.8) (0.21)

MangaJore 7.97 9.08 4.2 4.12 5.25 0.37

(4.62) (4.82) (3.4) (3.27) (3.93) (0.23)

Chennai 8.70 9.98 5.4 5.20 4.35 0.26

(5.14) (5.40) (4.0) (3.88) (3.25) (0.20)

Goa 7.27 8.18 4.9 4.76 4.86 0.22

(4.84) (5.46) (3.0) (2.86) (2.67) (0.17)

Hyderabad 15.53 16.24 11.3 10.92 11.11 0.52 (5.90) (5.91) (5.6) (5.53) (5.55) (0.29)

Vishakhapatnam 8.96 9.81 7.2 6.92 6.90 0.26

(6.98) (7.41 ) (5.6) (5.47) (5.44) (0.20)

Mumbai 10.81 12.07 6.0 5.45 6.20 0.18 (5.38) (5.77) (3.9) (3.83) (3.99) (0.13)

Kolkata 11.21 12.44 5.9 5.06 5.70 5.70 (5.9) (6.2) (4.1) (4.0) (4.0) (4.0)

Ahmedabad 7.62 8.58 4.8 4.8 5.00 5.00 (5.6) (5.8) (3.7) (3.9) (3.9) (3.9)

Bhopal 7.54 8.07 6.18 6.14 6.03 6.03

(6.71) (6.91) (4.5) (4.47) (4.58) (4.58)

Guwahati 11.87 12.92 6.1 4.3 5.90 5.90 (5.7) (5.9) (4.1) (3.7) (4.0) (4.0)

Jodhpur 14.32 15.51 9.8 9.73 9.97 9.97 (8.45) (8.63) (8.3) (8.50) (8.06) (8.06)

Lucknow 14.08 15.26 9.28 9.27 9.54 9.54

(13.33) (13.53) (14.2) (13.08) (13.00) (13.00)

Delhi 9.18 9.96 5.0 4.0 4.90 4.90 (6.7) (6.9) (3.9) (3.1) (3.9) (3.9)

Patiala 4.43 5.13 7.35 5.88 5.23 5.23

(3.29) (3.50) (4.7) (4.06) (3.75) (3.75)

Srinagar 5.49 4.78 2.0 2.1 1.97 1.97 (3.8) (3.8) (1.9) (1.8) (1.8) (1.8)

All stations 9.89 10.85 4.75 5.81 6.10 6.10 together (6.64) (6.86) (3.74) (5.43) (5.33) (5.33)

chayter 3 98

mainly based on the meteorological data from mid and high latitude regions. All the models

for ZHD show poor performance for the stations Bhopal, Jodhpur, Lucknow and Patiala

could be due to poor radiosonde data quality at these stations. Excluding these, the mean

absolute difference for ZHD estimated using Surface model (- 2.1 cm) is less than the other

two models (- 2.5 cm).

3.5 Variability in Zenith Tropospheric Delay and its Deviation from

Model Derived Values for Different Stations

The dry and wet component of the tropospheric zenith delay shows a rather systematic

annual variation, though the amplitude and phase show a variation with latitude. It would be

interesting in this context to examine the distribution of these delays on a day-to-day basis

for a period of one year to assess the extent of variability. Values of ZHD, ZWD and ZTD

(estimated through ray tracing) on different days are grouped at small intervals and the

percentage of occurrence of each group is estimated. Similarly, in order to examine the

model accuracies, the deviation of model predicted values of ZHD, ZWD and ZTD from the

true values estimated as above are also estimated for different days in a year and the

frequency distribution of these deviations are also examined. Figure 3.8 shows the

distribution of true ZHD obtained through ray-tracing and dZHD (true value of ZHD - ZHD

derived from Unified model). Figure 3.8a and Figure 3.8b shows the plots of these

distributions for different stations. Similarly Figure 3.9a and Figure 3.9b shows the

distribution of ZWD and dZWD for these stations, while Figure 3.lOa and Figure 3.l0b

shows the respective distributions of ZTD and dZTD. The distribution of ZHD is more-or­

less similar for all these stations except for Srinagar (for which it is truncated in the upper

half). The distribution is slightly skewed in its lower half for Bangalore while it is skewed in

its upper half for Bhopal. The distribution of model deviation for the ZHD is generally sharp

and symmetric for Ahmedabad and station south of it. For station north of it, the distribution

peaks at the highest value. The distribution of ZWD is more-or-Iess symmetric for

Bangalore, Portblair and Trivandrum. For other stations the distribution is rather flat

indicating that all values are equally probable. For stations like Mangalore,

Vishakhapatnarn, Mumbai and Delhi, the distribution shows a double hump feature. The

distribution of the deviation of model from true value of ZWD is more or less symmetric for

w !I! ;;! ~

u is .. w

U ! ~ u is .. ~ ~ ..

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(cm

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re 3

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: F

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odel

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atio

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(cm

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(cm

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Figu

re 3

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n o

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nith

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rost

atic

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(ZH

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ra

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me

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ba

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pal,

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wa

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ur,

lu

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, D

elhi

, P

atla

la,

and

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w - 8

w ~ g ... w

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TR

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(cm

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Fig

ure

3.9

a:

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quen

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n o

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nith

We

t D

elay

(Z

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) e

stim

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r T

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lr,

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galo

re,

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, G

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ha

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re

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requ

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00

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10

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ZT

D (

.. m

) ....

ZT

O (

cm

) Z

TO

(cm

) A

ZT

D (

cm

) Z

TD

(cm

) (\

ZT

D I

cm)

g,.,

l..g - :)- w

Figu

re

3.10

b:

Fre

quen

cy d

istr

ibu

tio

n o

f tr

ue

Ze

nith

Tro

posp

heri

c D

elay

(Z

TO

) e

stim

ate

d t

hro

ug

h

ray-

tra

cin

g

alon

g w

ith

th

e

dis

trib

utio

n o

f th

e m

odel

de

via

tion

s fr

om

th

e t

rue

val

ue f

or

Kol

kata

, A

hmed

abad

, B

hopa

l, G

uw

ah

atl,

Jo

dh

pu

r, L

uckn

ow,

Del

hi, I ~

Pat

lala

, an

d S

rin

ag

ar

105

Srinagar, Delhi, Guwahati, Ahmedabad and Kolkata. For other stations the distribution is

rather broad or non-symmetric. The distribution of ZTD is fairly symmetric for Srinagar,

Ahmedabad, Bangalore, Portblair and Trivandrum while Delhi; it also shows a double hump

like feature. For other stations it is rather skewed in the upper half. The model deviation of

ZTD is fairly systematic for all stations and is more governed by the shape of the ~WD

frequency distribution.

3.6 Summary

The neutral atmosphere introduces a delay in the propagation of microwaves, which

leads to an error for ranging in GPS based aircraft navigation, significant mainly during

landing and take-off. The neutral atmosphere being non-dispersive at GPS signal

frequencies, estimation of tropospheric delay is possible only through modeling. Simple

linear relationships are established for ZHD in terms of surface pressure and for ZWD in

terms of three humidity parameters; the surface water vapor pressure, surface water vapor

density and precipitable water vapor based on data from eighteen locations over the Indian

subcontinent as well as that from an Island station in the Bay of Bengal representative of

different climatic zones. In addition to these linear models based on meteorological data the

"characteristic height" parameters are modeled for different stations to implement the

Hopfield model. Using this model, the prediction of tropospheric error could be

accomplished for any given altitude above the surface. Taking note of the relatively small

variability of the coefficients in the above Site-Specific models, applicability of a Unified

model for the entire region encompassing these stations is evolved by pooling the data from

different stations. A ftrst order model in tenns of surface pressure for ZHD and a second

order nonlinear model for ZWD in tenns of water vapor partial pressure at the surface are

found to be most suitable for the prediction purpose. Following a similar approach unified

models are also developed for the Hopfield characteristic height parameters. All these

models are validated by comparing the model predictions with the true range errors obtained

through ray tracing the refractivity profiles derived from radiosonde data. As far as Site­

Specific models are concerned minimum deviation for the model prediction of ZHD is

observed at Trivandrum (- 0.17 cm) while the maximum deviation is observed for Srinagar

(- 2 cm). The validity of the Unified Surface model is also examined by comparing modeled

delay values with true ray-traced values as well as with other global models. In the case of

chayt(f 3 106

ZWD the deviation (- 5.81 ± 5.43 cm) of Unified Surface model is comparable to that of

site-specific model. The Unified surface model is found to be superior to the global models

when applied to Indian region.

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